A classical communication chain, illustrated in FIG. 1, comprises, for coding the signals coming from a source S, a source coder 1 (SCOD) followed by a channel coder 2 (CCOD) and, after the transmission of the coded signals thus obtained through a channel 3, a channel decoder 4 (CDEC) and a source decoder 5 (SDEC). The decoded signals are intended to be sent towards a receiver. Variable-length codes (VLC) are classically used in source coding for their compression capabilities, and the associated channel coding techniques combat the effects of the real transmission channel (such as fading, noise, etc.). However, since source coding is intended to remove redundancy and channel coding to re-introduce it, it has been investigated how to efficiently coordinate these techniques in order to improve the overall system while keeping the complexity at an acceptable level.
Among the solutions proposed in such an approach, the variable-length error correcting (VLEC) codes present the advantage to be variable-length while providing error correction capabilities, but building these codes is rather time consuming for short alphabets (and become even prohibitive for higher length alphabets sources), and the construction complexity is also a drawback, as it will be seen.
First, some definitions and properties of the classical VLC must be recalled. A code C is a set of S codewords {c1, c2, C3, . . . , ci, . . . cS}, for each of which a length li=|ci| is defined, with l1≦l2≦l3’ . . . ≦li≦ . . . ≦lS without any loss of generality. The number of different codeword lengths in the code C is called R, with obviously R≦S, and these lengths are denoted as L1, L2, L3, . . . , Li, . . . LR, with L1<L2<L3< . . . <LR. A variable-length code, or VLC, is then the structure denoted by (s1@L1, s2@L2, s3@L3, . . . , sR@LR), which corresponds to s1 codewords of length L1, s2 codewords of length L2, s3 codewords of length L3, . . . , and sR codewords of length LR. When using a VLC, the compression efficiency, for a given source, is related to the number of bits necessary to transmit symbols from said source. The measure used to estimate this efficiency is often the average length AL of the code (i.e. the average number of bits needed to transmit a word), said average length being given, when each symbol ai is mapped to the codeword ci, by the following relation (1):
                    AL        =                              ∑                          i              =              1                                      i              =              s                                ⁢                                    l              i                        ·                          P              ⁡                              (                                  a                  i                                )                                                                        (        1        )            which is equivalent to the relation (2):
                    AL        =                              ∑                          i              =              1                        R                    ⁢                                    L              i                        ·                          (                                                ∑                                      j                    =                                                                  r                        ⁡                                                  (                          i                          )                                                                    +                      1                                                                            j                    =                                          r                      ⁡                                              (                                                  i                          +                          1                                                )                                                                                            ⁢                                  P                  ⁡                                      (                                          a                      i                                        )                                                              )                                                          (        2        )            where, for a data source A, the S source symbols are denoted by {a1, a2, a3, . . . , as} and P(ai) is the respective probability of occurrence of each of these symbols, with ΣP(ai)=1 (from i=1 to i=S). If ALmin denotes the minimal value for the average length AL, it is easy to see that when ALmin is reached, the symbols are indexed in such a way that P(a1)≧P(a2)≧P(a3)≧. . . ≧P(ai)≧. . . P(as). In order to encode the data in such a way that the receiver can decode the coded information, the VLC must satisfy the following properties: to be non-singular (all the codewords are distinct, i.e. no more than one source symbol is allocated to one codeword) and to be uniquely decodable (i.e. it is possible to map any string of codewords unambiguously back to the correct source symbols, without any error).
An introduction and a presentation of different distances that are useful when reviewing some general properties of the VLC codes will then help to recall the notion of error-correcting property used in the VLEC code theory:
(a) Hamming weight and distance: if w is a word of length n with w=(w1, w2, . . . , wn), the Hamming weight of w, or simply weight, is the number W(w) of non-zero symbols in w:
                              W          ⁡                      (            w            )                          =                              ∑                          i              =              1                                      i              =              n                                ⁢                                    w              i                                                                    w                i                                                                                      (        3        )            and, if w1 and w2 are two words of equal length n with wi=(wi1, wi2, wi3, . . . , win) and i=1 or 2, the Hamming distance (or, simply, distance) between w1 and w2 is the number of positions in which w1 and w2 differ (for example, for the binary case, it is easy to see that:H(w1, w2)=W(w1+w2)  (4)where the addition is modulo-2). However, the Hamming distance is by definition restricted to fixed-length codes, and other definitions will be defined before considering VLEC codes.
(b) let fi=w1i w2i, . . . wni be a concatenation of n words of a VLEC code C, then the set FN={fi:|fi|=N} is called the extended code of C of order N.
(c) minimum block distance and overall minimum block distance: the minimum block distance bk associated to the codeword length Lk of a VLEC code C is defined as the minimum Hamming distance between all distinct codewords of C with the same length Lk:bk=min {H(ci, cj): ci, cjε C, i|ci|=|c j|=Lk} for k=1, . . . ,R  (5)and the overall minimum block distance bmin of said VLEC code C, which is the minimum block distance value for every possible length Lk, is defined by:bmin=min{bk: k=1, . . . R}  (6)                (d) diverging distance and minimum diverging distance: the diverging distance between two codewords of different length        
      c    i    =                    x                  i          1                    ⁢              x                  i          2                    ⁢                          ⁢      …      ⁢                          ⁢              x                  i                      l            i                              ⁢                          ⁢      and      ⁢                          ⁢              c        j              =                  x                  j          1                    ⁢              x                  j          2                    ⁢                          ⁢      …      ⁢                          ⁢              x                  j                      l            i                              of a VLEC code C, where ci, cj ε C, li=|cj| and lj=|cj| with li>lj, is defined by:
                              D          ⁡                      (                                          c                i                            ,                              c                j                                      )                          =                  H          ⁡                      (                                                            x                                      i                    1                                                  ⁢                                  x                                      i                    2                                                  ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  x                                      i                                          l                      i                                                                                  ,                                                x                                      j                    1                                                  ⁢                                  x                                      j                    2                                                  ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  x                                      j                                          l                      j                                                                                            )                                              (        7        )            i.e. it is also the Hamming distance between a lj-length codeword and the lj-length prefix of a longer codeword, and the minimum diverging distance dmin of said VLEC code C is the minimum value of all the diverging distances between all possible couples of codewords of C of unequal length:dmin=min{D(ci,cj):cicj ε C,|ci|≠|cj|}  (8)                (e) converging distance and minimum converging distance: the converging distance between two codewords of different length        
      c    i    =                    x                  i          1                    ⁢              x                  i          2                    ⁢                          ⁢      …      ⁢                          ⁢              x                  i                      l            i                              ⁢                          ⁢      and      ⁢                          ⁢              c        j              =                  x                  j          1                    ⁢              x                  j          2                    ⁢                          ⁢      …      ⁢                          ⁢              x                  j                      l            i                              of a VLEC code C, where |ci|=li>|cj|=lj, is defined by:
                              C          ⁡                      (                                          c                i                            ,                              c                j                                      )                          =                  H          ⁡                      (                                                            x                                      i                                                                  l                        i                                            -                                              l                                                  j                          +                          1                                                                                                                    ⁢                                  x                                      i                                                                  l                        i                                            -                                              l                                                  j                          +                          2                                                                                                                    ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  x                                      i                                          l                      i                                                                                  ,                                                x                                      j                    1                                                  ⁢                                  x                                      j                    2                                                  ⁢                                                                  ⁢                …                ⁢                                                                  ⁢                                  x                                      i                                          l                      j                                                                                            )                                              (        9        )            i.e. it is also the Hamming distance between a lj-length codeword and the lj-length suffix of a longer codeword, and the minimum converging distance of said VLEC code C is the minimum value of all the converging distances between all possible couples of C of unequal length:cmin=min{C(ci,cj): ci, cj ε C,|ci|≠|cj|}  (10)                (f) free distance: the free distance dfree of a code is the minimum Hamming distance in the set of all arbitrary long paths that diverge from some common state Si and converge again in another common state Sj, with j>i:dfree=min{H(fi, fj): fi, fj ε FN, N=1, 2, . . . , ∞}  (11)        
Following the structure model used for a VLC, it is therefore possible to describe the structure of the VLEC code C by the notation:S1@ L1, b1; S2 @ L2, b2; . . . ; SR@ LR, bR; dmin, cmin  (12)where there are si codewords of length Li with minimum block distance bi, for all i=1, 2, . . . R, (it is recalled that R is the number of different codeword lengths) and minimum diverging and converging distances dmin and cmin. The most important parameter of a VLEC code is its free distance dfree, which influences greatly its performance in terms of error-correcting capabilities, and it can be shown that the free distance of a VLEC code is bounded by:dfree≧min(bmin, dmin+cmin)  (13)
These definitions being recalled, the state-of-the-art in VLEC codes construction will be now described more easily. The first types of VLEC codes, called α-prompt codes and introduced in 1974, and an extension of this family, called at1,t2, . . . ,tR-prompt codes, have both the same essential property: if one denotes by α(ci) the set of words that are closer to ci than to any codeword cj, with j≠i, no sequence in α(ci) is a prefix of a sequence in another α(ci). The construction of these codes is very simple, and the construction algorithm is adjustable by the number of codewords at each length, which makes possible to find the best prompt code for a given source and a given dfree. However, this best code performs poorly in terms of compression performance.
A more recent construction, allowing the construction of a VLEC code from the generator matrix of a fixed-length linear block code, was proposed in the document “Variable-length error-correcting codes” by V. Buttigieg, Ph.D. Thesis, University of Manchester, England, 1995. Called code-anticode construction, this algorithm relies on line combinations and column permutations to form an anticode at the rightmost column. Once the code-anticode generator matrix is obtained, the VLEC code is simply obtained by a matrix multiplication.
This technique has however several drawbacks. First, there is no explicit method to find the needed line combinations and column permutations to obtain the anticode. Moreover, the construction does not take into account the source statistics and, consequently, often reveals itself sub-optimal (one can find a code with smaller average length by a post-processing on the VLEC code). In the same document, the author has then proposed an improved method, called Heuristic method, that is based on a computer search for building a VLEC code giving the better known compression rate for a specified source and a given protection against errors, i.e. a code C with specified overall minimum block, diverging and converging distances (and hence a minimum value for due) and with codeword lengths matched to the source statistics so as to obtain a minimum average codeword length for the chosen free distance and the specified source (in practice, one takes: bmin=dmin+cmin=dfree, and: dmin=[dfree/2].
The main steps of this Heuristic method, which uses the following parameters: minimum length L1 of codewords, maximum length Lmax of codewords, free distance dfree between each codeword, number S of codewords required, are now described with reference to the flowcharts of FIGS. 2 to 4.
To start the computer search (“Start”), all the needed parameters must be first specified: L1 (the minimum codeword length, which must be at least equal to or greater than the minimum diverging distance required), Lmax (the maximum codeword length), the different distances between codewords (dfree, bmin, dmin, cmin), and S (the number of codewords required by the given source), and some relations are set when choosing these parameters:
Ll≦dmin 
bmin=dfree 
dmin+cmin=dfree 
The first phase of the algorithm, referenced 11, is then performed: it consists in the generation of a fixed length code (put initially in C) of length L1 and minimal distance bmin, with a maximum number of codewords. This phase is in fact an initialization, performed for instance by means of an algorithm such as the greedy algorithm (GA), presented in FIG. 5, or the majority voting algorithm (MVA), presented in FIG. 7, or a new proposed variation, denoted by GAS (Greedy Algorithm by Step), which consists in a variation of the two above mentioned ones (the GAS consists in the search method used in the GA, where instead of deleting half of the codewords, only the last codeword of the group is deleted). These two algorithms are useful to create a set W of n-bit long words distant of d (in practice, it may be noted that the MVA finds more words than the GA, but it asks too much time for only a small improvement of the compression capacity, as shown in the tables of FIGS. 6 and 8, which compare, respectively for the GA and for the MVA, the best code structures obtained with different values of dfree for the 26-symbol English source defined in the table of FIG. 9.
The second phase of the algorithm, corresponding to the elements referenced 21 to 24 (21+22=operation “A0”; 23+24=operation “A2”) in FIG. 2, consists in listing and storing (step 21) in a set called W all the possible L1-tuples at the distance of dmin from the codewords in C. If dmin≧bmin, then W is empty. If this set W of all the words satisfying the minimum diverging distance to the current code is not empty (reply NO to the test 22: |W|=0 ?), the number of words in W is doubled by increasing the length of the words by one bit by affixing first a “0” and then a “1” to the rightmost position of all the words in W (step 24), except if the maximum number of bits is exceeded (reply YES to the test 23). At the output of said step 24, this modified set W has twice more words than the previous W, and the length of each one is L1+1.
The third phase of the algorithm, corresponding to the elements 31 to 35 (=operation “A3” in FIG. 2), consists in deleting (step 31) all the words of set W that do not satisfy the cmin distance (minimum converging distance) with all the codewords of C (i.e. in keeping and storing in a new W only the words which satisfy said minimum converging distance, the other ones being discarded). At this point, the new set W is a set of words which, when compared to the codewords of C, satisfy the required minimum diverging and converging distances (both dmin and cmin distances) with the codewords of C. If that new set W is not empty (reply NO to the test 32: |W|=0 ?) one selects in W (step 33) the maximum number of words to satisfy the minimum block distance, in order to ensure that all the words of the set W, being of the same length, have a minimum distance at least equal to bmin. At the end of this step 33, realized with the GA or the MVA (note that in this case, the initial set used for the GA or the MVA is the current W and not a n-tuples set), the words thus obtained are added (step 34) to the codewords already in C.
If no word is found (i.e. W is empty) at the end of the step 21 (reply YES to the test 22: |w|=0 ?) or if the maximum number of bits is reached or exceeded (reply YES to the test 23), one enters the fourth phase of the algorithm (steps 41 to 46, illustrated in FIG. 3 and also designated by the operation “A1” in said figure), which is used in order to unjam the process by inserting more liberty of choice, more particularly by affixing to all words in W extra bits (several bits at the same time) such that the new group contains more bits than the old one. If there are enough codewords in the last group (successive tests 41 and 42, for verifying the number of codewords in the last group, and if there are previous groups), some of them are deleted from this said group (as described above), such deletions allowing to reduce the distance constraint and to find more codewords than before. As a matter of fact, the classical Heuristic method thus described begins with the maximum of codewords with the short length, maps them with the high probability symbols and tries to obtain a good compression rate, but sometimes the size of the small lengths sets are incompatible with the required number of codewords S. In this optic, easing a few codewords provides more freedom degrees and allows to reach a position where the initial requirements on distance and number of symbols for the code can be met. This deletion process is repeated until it remains a maximum of one codeword for each length. If W is empty at the end of the step 31 (reply YES to the test 32: |W|=0 ?), the steps 23, 24, 31, 32 are repeated. If the required number of codewords has not been reached (reply NO to the test 35 provided at the end of this third phase), the steps 21 to 24 and 31 to 35 must be repeated until said steps find that either there are no further possible words to be found or the required number of codewords is reached.
If said required number of codewords has been reached (i.e. the number of codewords of C is equal to or greater than S (reply YES to the test 35), the structure of the VLEC code thus obtained is used in a fifth part, including the steps 51 to 56 (illustrated in FIG. 4, and also designated by the operation “A4” in said figure), in order to calculate the average length AL. This is done by weighting each codeword length with the probability of the source, and comparing it to the current best one. If said average length AL of this VLEC code is lower than the minimized value of AL (=ALmin), this AL becomes the ALmin, and this new AL value and the corresponding code structure are kept in the memory (step 51). These steps 51 and following (fifth part; operation “A4”) allow to come back, within the algorithm, towards previous groups, while the other phases of said algorithm are always performed on the current group. The stepsize for such a feedback operation is one, i.e. this feedback action can be considered as exhaustive.
To continue this search of the best VLEC code, it is necessary to avoid keeping the same structure, which would lead to a loop in the algorithm. The last added group of the current code is deleted (steps 52, 53), the deletion of shorter length codewords allowing to find more longer length codewords (test 54: number of codewords in group greater than 1 ?), and some codewords (half the amount for the GVA; the “best” one for the MVA) of the previous group are deleted (step 55), in order to re-loop (step 56) the algorithm at the beginning of the step 21 (see FIG. 2) and find different VLEC structures (the number of deleted codewords depends on which method is used for selecting the words: if the GA method is used and one wants to obtain a linear code, it is necessary to delete half of the codewords, while with the MVA method only one codeword, the best one, is deleted, i.e. the one that allows to find the more codewords in the next group).
However, the Heuristic method thus described often considers very unlikely code structures or proceeds with such a care (in order not to miss anything) that a great complexity is observed in the implementation of said method, which moreover is rather time consuming and can thus become prohibitive.